Question
How to select superior soybean genotypes across locations and years (GxE interaction) according to Multitrait ideotype? (Simultaneous selection)
Hypothesis
Estimate probability of superior performance (Dias et al. 2022) across seasons and years and classify genotypes using Bayesian Probabilistic Selection Index (Chagas et al. 2025)
Important
Select superior soybean genotypes to grain yield, plant height and plant lodging using Bayesian probabilistic selection index (BPSI) (Chagas et al. 2025)
Individual analyses
\[ \mathbf{y} = \mathbf{X_1b} + \mathbf{Z_1g} + \mathbf{\epsilon} \]
onde \(\mathbf{y}\) é o vetor das observações fenotípicas, \(\mathbf{b}\) é o vetor dos efeitos fixos de repetição, \(\mathbf{g}\) é o vetor dos efeitos aleatórios de genótipo e \(\mathbf{\epsilon}\) é o vetor dos efeitos residuais. \(\mathbf{X_1}\), \(\mathbf{X_2}\) e \(\mathbf{Z_1}\) são matrizes de incidência dos efeitos \(\mathbf{b}\) e \(\mathbf{g}\) respectivamente.
\[ h^2 = \sigma^2g / \sigma^2g + \sigma^2e \]
onde \(\sigma^2g\) é a variância genética e \(\sigma_e^2\) é a variância do erro.
\[ CV = \frac{\sigma_e}{\mu} \times 100 \]
onde \(\mu\) é a média da característica.
\[LRT= −2 \times (Log𝐿 - Log L_𝑅)\]
onde \(L\) é o ponto máximo da função de verossimilhança restrita do modelo completo e \(L_R\) é o mesmo para o modelo reduzido, ou seja, sem o efeito a ser testado. O valor de LRT foi comparado com o valor tabulado com base na tabela qui-quadrado, a um grau de liberdade e probabilidade de 0,95.
\[ y_{jkhp} = \mu + t_h + l_k + b_{p(k)} + g_j + gl_{jk} + gt_{jh} + \varepsilon_{jkhp} \] where the \(y_{jkhp}\) is the phenotypic record of the \(j^{th}\) genotype, allocated in the $ p^{th}$block, in the \(k^{th}\) location and in the year \(t_{th}\). All other effects were previously defined but \(b_{p(k)}\), which is the effect of the \(p^{th}\) block in the \(k{th}\) location, and \(gl^{jk}\) , which correspond to the genotype-by-location interaction \(t_h\) and \(t_{jh}\) are the main effect of years and the genotypes-by-years interaction effect, respectively.
BPSI index uses the probability of superior performance to estimate the chance of a genotype being selected in multienvironmental trials (Dias et al. 2022).
\[ Pr\left({\hat{g}}_i \in \Omega \middle| y\right) = \frac{1}{s}\sum_{s=1}^{s} I \left({\hat{g}}_i^{(s)} \in \Omega \middle| y\right) \]
where \(\hat{g}_i\) is the genotypic value, \(\Omega\) is a subset of genotypes with superior performance and \(s\) represents each sample of posterior distribution.
\[ BPSI_i = \sum_{m=1}^{t} \frac{RankProbSup^t}{\omega^t} \]
where \(t\) is the total number of traits evaluated \((m =1, 2,…,t)\) and \(\omega\) is a weight. Traits of greater interest will have larger \(\omega\). We used weight 2 for GY and weight 1 for PH and PL. The 10% best-ranked families were selected according to the BPSI.
| Genotype | T1(Rank) | T2(Rank) | T3(Rank) | PSI |
|---|---|---|---|---|
| 1 | 10 | 5 | 2 | ∑ i.= 17 |
| 2 | 5 | 3 | 10 | ∑ i.= 18 |
| 3 | 7 | 3 | 10 | ∑ i.= 19 |
DGM Lab